A robust sliding mode control for pneumatic servo systems

Pergamon

Int. J. Engng Sci. Vol.35, No. 8. pp. 711-723,1997 ~) 1997ElsevierScienceLimited.All rightsreserved Printed in GreatBritain PII: S0020-7225(96)00124-3 0020-7225/97 $17.00+0.00

A ROBUST SLIDING MODE CONTROL FOR PNEUMATIC SERVO SYSTEMS J U N B O S O N G and Y O S H I H I S A I S H I D A Department of Electronics and Communication, Facultyof Scienceand Technology,Meiji University,1-1-1, Higashi-mita,Tama-ku, Kawasaki, 214 Japan (Communicated by S. MINAGAWA) Abstraet~This paper presents a robust sliding mode control scheme for pneumatic serve systems, By using the Lyapunov stability theory and a few structural properties of pneumatic serve systems, a robust sliding mode controller can be designed so that the output tracking error within any neighborhood of boundary layer as time, t--*~, and strong robustness, with respect to large uncertain dynamics, can be guaranteed. The pneumatic serve systems using a proposed control scheme have a strong, robust property, not only because the error dynamic is insensitive to uncertain dynamicson the sliding mode, hut also because only the uncertain bounds based on the structure properties of pneumatic serve systems are used in controller design. Its superior performance is suggested in the results, both simulation and application in the practical pneumatic serve system~ C) 1997 Elsevier Science Ltd. 1, I N T R O D U C T I O N As an important driving element, the pneumatic cylinder is widely used in industrial applications because of its reliable, cheap and excellent performance in an industrial serve system. However, along with the development of control technology the requirement for control precision gets higher and higher, as well as the structure of pneumatic cylinders becoming more and more complex. In many cases, in order to achieve the satisfied control performance, we have to consider the effects of nonlinear factors contained in the plant. In recent years, in order to improve the control performance, several methods, such as feedback linearJLzation, observer design, adaptive control, singular perturbation and fuzzy control, have been investigated in order to design effective control for different industrial control systems, such as robots, motors and pneumatic serve systems etc. A list of references, showing these results, can be found in the survey paper [1]. We shall only give a brief synopsis of the most recent developments and provide a comparison of different approaches. In the feedback linearization method [2, 3], control is designed based on feedback linearization transformation, which requires the knowledge of the dynamics and acceleration measurements. Under the assumption that the nonlinear uncertainty is small, there are two time scales in system dynamics and a control can be designed using a singular perturbation technique. Some classical adaptive control for robots was studied in [4--6]. The resulting control is intuitively simple, since it usually contains two parts: a control for rigid body and a corrective control. The control, however, does not guarantee global stability. The most recent adaptive control scheme for a pneumatic serve system, proposed in [7], described an adaptive poles placement control. Since it avoided cancelling the unstable zero of the plant, the robustness was an improvement compared to other application studies~ However, it seems that the effectiveness of nonlinear uncertainties caused by air compression and friction dynamics were omitted. An observer design technique for a pneumatic serve system is shown in [8], it rejects the offset caused by stick friction and the external disturbance dependent on an external disturbance observer, which includes an inverse system, so it is necessary to know all about or have a part knowledge of plants. The fuzzy control was used to solve nonlinear factors [9], it is based on the classical PI control and combined with fuzzy theory to improve the robustness of the control system. 711

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J. SONG and Y. ISHIDA

The robust control approach to model tracking has been a very active area of research. Such as Qu's series model reference robust control (MRRC) [10-13], they show the controller design method from minmax Lyapunov design for continuous control system. Qu's results give us the capability of handling the control problem as MRAC using robust control theory. In particular, robust control for a practical industrial system has attracted much attention from control researchers [14-16]. The main reason is shown by experimental study [17-19], where, in a number of good behavior control systems, the nonlinear uncertainty factors contained in the plant must be taken into account in the design of a controller, in order to achieve good control performance. In this paper, we develop a robust controller for pneumatic servo systems using sliding mode technique and a few structural properties of pneumatic servo systems. It can then be designed so that the output tracking error cannot become any larger than an arbitrary small constant, as time, t---~0o, and strong robustness, with respect to large uncertain dynamics, can be guaranteed. This paper is organized as follows. In Section 2 we formulate our problem, the solution of which is presented in Section 3. The simulation and practical application of experimental results are shown in Sections 4 and 5. Finally, in Section 6 we draw some conclusions of this work.

2. P R O B L E M STATEMENT First, we consider the pneumatic cylinder in the servo systems shown in Fig. 1. If the nonlinear uncertainty factors can be omitted, the dynamics of the pneumatic cylinder can be described by the following second-order linear differential equation d2y

M-~

dy

+ ~7--~ + ky= u(t)

(1)

where y is the position of piston, u(t) is the internal pressure of the pneumatic cylinder, M is the load mass, ~/ is the damping coefficient and k is the spring constant. However, in a practical system, the friction and air compression cannot be described as strictly linear functions of velocity anti position. In this case, we have to take these nonlinear factors into account for a good behavior servo control. Considering the operating property of the pneumatic cylinder, the dynamics of the pneumatic cylinder can be modeled as d2y

M(y,t) - ~

+ Ff(M,)~) + Fp(y,y) = u(t) + d(t)

(2)

where Ff(M,p) is the friction dynamics, Fp(y,)~) is caused by air compression and d(t) is the external disturbance. The nonlinear functions M(y,t), Ff(M,p) and Fp(y,y) are unknown.

I

,

Pneumatic Cylinder Fig. 1. Analysismodel of the pneumatic cylinder.

Robust sliding mode controller for pneumatic servo systems

713

Further, we assume that the u(t),y and ) are measurable. For simple analysis, equation (2) can be rewritten as d2y

M(y,t) - ~

+ F(M,y,y) = u(t) + d(t)

(3)

where F(M,y,y) = Ff(M,y) + Fp(y,y). If we define the state vector x as x = [y,p]T

(4)

Q(y,t) = M-~(y,t)

(5)

and let

the plant described in equation (3) can be described as

' ]+[~]Q(y,t)[u(t)+d(t)]. i = I -S Q(y,t)F(M,y,y)

(6)

The reference model for the following plant can be represented as follows:

[]:[ Ym

0

Ym

-- a m l

][] [0]

1

Ym +

-- am2

Ym

r(t)

=

A mXm + Bmr(t)

(7)

bm

where the parameters am~, am2 and b,, are the positive constant, which are chosen such that the reference model equation (7) is stable. The reference input signal r(t) and output Ym of equation (7) and its differential signal )~m are assumed to be measurable. For further analysis, the following uncertain bounds for the pneumatic servo system, defined in equation (3), are assumed to be known.

ql <- Q(y,t) <- qz

(8a)

IF(M,y,y)l <- fflyl + folyl + fs Id(t)l < Ip(/)l

(8b) (8c)

where the parameters ql, q2, ff, fp and fs are positive numbers and p(t) is a known function which is selected such that it make equation (8a) tenable. REMARK 2.1. According to the operation characteristics of pneumatic servo system, the model equation (3) is reasonable. M(y,t) can be thought of as the load mass, which is variable in a different operation position. F(M,y,S,) is the function connecting the friction dynamics model [20] and a practical air compression procession. Based on the main properties of pneumatics cylinders, the assumptions equations (6)-(8a) can be explained as tollows. REMARK 2.2. Assumption equation (8a) implies that the variations of load mass must be within the range cylinder allowed and then the mass of the piston is not zero. REMARK 2.3. Because the variant range of load mass is not infinite, moreover, the velocity of cylinder's piston always has a finite value and then the air pressure is finite. Hence, assumption equation (8b) is ]reasonable for the pneumatic servo system. REMARK 2.4. Usually, the random external disturbance has a small power and amplitude and the regular disturbance is always found with some varying rules, which can be used to synthesize the bounded function p(t). The objective of this paper is to design a robust sliding mode controller based on uncertain bounds in assumptions equations (8), instead of the upper and the lower of all unknown system parameters, so that for any bounded uncertainties in parameters of pneumatic servo systems,

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J. SONG and Y. ISHIDA

U(t )_[ Pneumatic I Y_r -I Cylinder I' ~

r ( t )dl

Reference -I~ .

-] model

Fig. 2. A pneumatic cylinder model and a desired reference model.

the tracking error between the plant and reference model described in equation (7) can be guaranteed within any neighborhood of the boundary layer as time t ~ oo.

3. R O B U S T S L I D I N G M O D E C O N T R O L L E R

3.1. Controller design Based on the definitions and assumptions in Section 2 we can define the system error equation as e = y - Ym

(9)

k = )) - 3~,.

(10)

e =

(11)

and then the error vector e as

Using the expressions equations (6) and (7), we obtain the error differential equation as = Ame + Bf(M,y,y,u,d,r, 0

(12)

where B = [ 0 , 1] T and

f(M,y,y,u,d,r,t) + Q(y,t)(u + d) + fl(M,y,p,r,t)

(13)

f l(M,y,y,r,t) = am~y + am2P - bmr - Q(y,t)F(M,y,y).

(14)

To design a robust sliding model control system, in which the error dynamic converges to zero as time t--.oo, we define the following sliding switching variable S = Ce

(15)

C = [cl,c2] = [c1,1], cl > 0.

(16)

where

REMARK 3.1. For the simplified analysis element c2, in expression equation (16), is chosen as 1, however, it can be chosen as another positive number for the improvement of the convergence of the error dynamics. THEOREM 3.1. Consider the nonlinear plant in expression equation (3) and the error dynamics in expression equation (12) with assumptions in equations (8). If the control law is designed such as 1

-

U(I)

0

S

q-T'~w(t)

for

ISl~0

for

ISI = 0

(17)

where

w(t) = [amlyml + la,,zp,,I + tbmrl + q~lpl + c~lkl + qz(ffl~91+ fplyl + fs) then the output tracking error converges to zero as t---~0.

(18)

Robust sliding mode controller for pneumatic servo systems

715

PROOF. Consider the following Lyapunov function 1 V = - - S 2. 2

(19)

Differentiating V with respect to time, we have ¢" = s s = s ( c ~ ).

Submitting the expressions equations (11)-(13) into equation (19) we obtain I2 = S[CA,,,e + C B f ( M , y , y , u , d , r , t ) ]

= S[CAme + Q ( y , t ) ( u + d) + f l ( M , y , y , r , t ) ]

= S[CAme + Q ( y , t ) ( u + d) + am~y + am2y - bmr - Q ( y , t ) F ( M , y , y ) ] = S[am~Ym + am2Ym + Q ( y , t ) ( u + d) - bmr + c t k - a ( y , t ) F ( M , y , p ) ] .

(20)

Let k , ( t ) = --1 Q ( y , t ) ql

(21)

and then consider assumption equation (8a); we have k l ( t ) > - 1 . Submit the control law equation (16) into expression (20), then expression (20) can be written as

(I = - [kzlSl(lamlyml + lam2)~ml)+ S(am~y,, + am2)~m)] - [k~lSl(Ibmrl + c~[kI) + S ( b m r - c~k )] - [k~lSIqz(fflyl + fplyl + f s ) + S Q ( y , t ) F ( M , y , p ) ]

- [ Q ( y , t ) ( I S p l - Sd)].

(22)

Considering k l ( t ) --- 1 and ISI ~ 0 , we have f" <- - [k~lSIqz(ffl)l + fplyl + f s ) + S Q ( y , t ) F ( M , y , y ) ]

= - tqlSl[q2(fflyl + fplyl + fs) +

Q(y,t) k~

- [ Q ( y , t ) ( I S p l - Sd)]

S ISI F ( M , y , y ) ] - [ Q ( y , t ) ( I S p l - Sd)]

= - kllSIq~(t) - [ Q ( y , t ) ( I S p l - Sd)]

(23)

where Q(y,t) ~b(t) = q2(ff[)l + fplyl + f , ) +

k,

S ISI F ( M , y , p ) >- q2

(ffl)[ + fplyl + f s ) - q a l a ( y , t ) l l F ( M , y , y ) l . (24) Using assumptions 8a, b, we have ~ ( t ) - 0 . Furthermore, considering assumption (8c) we have Ipl - d>0.

(25)

Then (z < - k~lSIdp(t) - [ Q ( y , t ) ( I S p l - Sd)] <- - kllSIdp(t) - [ Q ( y , t ) ( I S p l - ISdl)] <- - k,ISIfb(t) - [ Q ( y , t ) l S l ( I p l - Idl)] <0.

(26)

It can be shown lhat S converges to zero as t--->oo. Considering the definition of S in equation (14), we have k + cle = 0. (27) If we let the initial value of s at t = 0 as e(0), we can obtain the solution of equation (27) as

e (t) = e (0) e - c,,.

(28)

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J. SONG and Y. ISHIDA

Obviously, expression equation (28) satisfies e ( t ~ O ) as t ~ tracking error. Thus, we complete the proof of Theorem 3.1.

for any initial value of the

3.2. Smoothing the control law We have proven that the output tracking error can converge to zero, as t---~~, by the use of the proposed control law in equation (17). However, this control law is discontinuous across the S = 0, which may excite undesired high frequency dynamics. This problem can be eliminated by smoothing out the discontinuous control law in the neighborhood of boundary layer. To do this, we replace signum nonlinearity by a saturation nonlinearity, which is defined as follows: 1 for S/8 for - 1 for

sat(S)=

S/8>1 -1-
(29)

where 8 > 0 is the boundary layer thickness. With this boundary, the control law resulting from Theorem 3.1 becomes 1 u(t) = - - - sat (S)w(t). ql

(30)

THEOREM 3.2. Consider the nonlinear plant in expression equation (3) and the error dynamics in expression equation (12) with assumptions in equations (8). If we choose the control law equation (30), the plant output tracking error is globally uniformly ultimately bounded. That is, as time approaches infinity, the magnitude of the output tracking error becomes no larger than a constant, 8, which can be set as arbitrarily small. PROOF. The proof is similar to that given by Theorem 3.1. We select Lyapunov function as

1S~ V = -~-

(31)

$8=S -sat(S)&

(32)

17 = $8S8 = $8C6.

(33)

where

Thus, we have

Using the control law equation (30) and submitting expressions equations (11)-(13) into equation (33), we obtain 17 = - [klSssat(S)(la,myml + la,.2)~.,I) + Ss(a.,~y,. + am2Ym)] - [klSssat (S)(Ib,.rl + c~lk I) + Ss(b.,r - c~k)] - [k,Sssat (S)q2f + SsQ(y,t)F(g,y,y)] - [Q(y,t)(ISssat(S)pl- Ssd)]

(34)

where kl = Q(y,t)/ql. According to the definitions of sat(S) and $8, in the case of ISl >8, we have Sssat (S) = 1581

(35)

$ 8 # 0.

(36)

and

Furthermore, using the result of equation (26) and considering the kl -> 1, we have I7 = S,~¢8.

Robust slidingmode controller for pneumaticservo systems

717

Noting that the ,definitions of sat(S) and $8 make the $8 = 0 for ISI -< 8. As well as considering the definitions o:f S and $8, it can be shown that the output tracking error lel ~ 8 as t ~ o0 for any initial e (0). ".['he proof of Theorem 3.2 has been completed. REMARK 3.2. The control resulting from Theorem 3.2 can guarantee global stability and robust performance. Although the output tracking error converges globally and exponentially to ultimate bound B, the tracking error is not guaranteed to converge to zero, despite the fact that the design parameter 8 > 0 can be made arbitrarily small if sufficient control energy is available. REMARK 3.3. We note that the tracking error can be made smaller by choosing smaller values for the design pa'mmeters 8. If 8 is too small, however, the output of the robust controller may change rapidly and have a large magnitude. Since the bounds and bounding functions are usually conservatiive, the design parameter, 8, should not be made too small during the transient phase and could gradually be made smaller if the output tracking error is larger than required.

4. S I M U L A T I O N E X A M P L E S To illustrate the control scheme proposed in this paper, a simulation example for a secondorder model conducted by a pneumatic cylinder is studied. The dynamic equation is given in [9,20] as

M(y,t)y + F(M,y,y) = 13.59[u(t) + d(y,t)] where

M(y,t) = 30 + 701 sin(y)l F(y,y,M) = (3.25 x M - 250 exp ( - lyl/20)) sign ( ) ) a ( y , t )

= 100 sin (t) sin (y)/(1 + lyl).

A reference model for a plant to follow is given by

and r(t) = sin(0.5cr 0 for t>0. Since we are interested in trajectory tracking and hope that the transient response is desired entirely by reference model, we consider a situation characterized by the same initial value of both the reference model state and the plant state. In this simulation, we pick up the initial values of plant and reference as zero. The parameters of the uncertain bounds and bound function in equations (8) are given by ql = 0.005, q2 = 0.1 f , = 4.0, fp = 0.5, f~ = 1000

p(t) = 500, cl = 2.0. Figure 3 shows the output tracking, tracking error and input signal by the use of the control law resulting from Theorem 3.1. It can be seen that the efforts of system uncertainties are eliminated and good tracking performance is achieved, as well as showing that the input signal is discontinuous~ To overcome this problem, we implemented the boundary layer control law

718

J. SONG and Y. ISHIDA

resulting from Theorem 3.2 and set 8 = 0.005. Satisfied system performance is shown in Fig. 4. As shown in these figures, the chattering is eliminated. 5. A P P L Y I N G T O A P R A C T I C A L

SERVO SYSTEM

Although the proposed control scheme has illustrated a good performance in the simulation example, in order to prove the applicability in practical pneumatic servo systems shown in Fig. 5, an application of the proposed control scheme is carried out in this study. Figure 5 shows the configuration of the pneumatic servo position control system. The system is primarily composed of the pneumatic cylinder, a 32-bit personal computer with an 80486 CPU 15 I , " ~(a)

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10

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c ¢/)

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0

5

10

15

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Time(See) Fig. 3. S i m u l a t i o n results o f the plant w i t h c o n t r o l law (17): (a) the o u t p u t tracking; (b) the o u t p u t t r a c k i n g error; (c) the c o n t r o l signal.

Robust sliding mode controller for pneumatic servo systems

719

and an associated arithmetic coprocessor, two 12-bit D/A converters, two servo amplifiers of the voltage-current transformation type and two electro-pneumatic proportional valves, The pneumatic cylinder (CA1BQ63-300:SMC Corporation) is a low friction type whose minimum pressure is 0.01 MPa, inside diameter is 634, and the stroke is 300 mm. A load mass 10 kg is always connected to the tip of a position rod. The electro-pneumatic proportional valve is a proportional control valve (VEP3120-1: SMC Corporation) in pressure being of 3 port. The position and velocity measurement is carried out by a magnetic sensor whose resolving power is 10/.tm (SR10-100A: SONY Corporation). We assume that the pneumatic cylinder in this system satisfies the assumptions in equations (8) and its dynamic motion equation can be described as equation (3). Obviously, it is impossible to apply the control law, equation (17), for this plant, because it is a zero sampling interval control law. In the following experiment, the control law, 1.5

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720

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p(t) = 500, 8 = 2, cl = 2.0 and select reference model as -64-

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In the experiment, the reference input signal r(t) = 5011(0 -

- 8t)].

According to the transporting operation's features of the pneumatic cylinder, the plant is put into different load masses in forward and backward directions. The experimental results are shown in Figs 6 and 7. Figure 6 shows the results where the load mass is 30 kg in the forward direction, 100 kg in the backward direction. Figure 7 shows the ones where the load mass is 100 kg in the forward direction, 30 kg in the backward direction. As shown in Figs 6 and 7 the dynamical tracking error is no larger than 2 mm and the static control precision is about 0.2 mm. Furthermore, the control signal is continuous, as is time. It can be seen that the effects of nonlinear uncertainty factors contained in the plant are restrained and good tracking performance is achieved. As pointed out in Remark 3.3, in the system shown in Fig. 5 the ~ cannot be selected too small, the experimental results show that the choice of 8's size seems dependent on the sampling interval. If we want to improve the tracking precision further, it is necessary to increase the sampling frequency. Although we cannot achieve arbitrary tracking precision in the pneumatic servo system with the proposed control scheme, because of the limit of sampling frequency, the experimental results still suggest that this control scheme is useful for a practical pneumatic servo system, if it can be modeled by expression equation (3) and satisfies the assumptions equations (8). 6. CONCLUSIONS In this paper, a robust sliding mode control scheme has been proposed. The main contribution of this paper is that it has found a robust controller (Theorem 3.1) for a pneumatic

Robust sliding mode controller for pneumatic servo systems

721

servo system which can guarantee that the tracking error converges to zero as time t--->~. Additionally, the pneumatic servo systems using the proposed scheme have a strong, robust property, not only because of the fact that on the sliding m o d e the error dynamics are insensitive to uncertain dynamics, but also because only the uncertain bounds and bound function are used in controller design. Like all other control techniques, the ideal error convergence cannot be obtained in practical control systems where the sampling interval is nonzero. To implement this scheme, we developed T h e o r e m 3.1 into Theorem 3.2, as it allows the setting of a suitable dead zone 6 to smooth the control signal. However, the control scheme proposed in the p a p e r can only be used in a second-order pneumatic servo system. The research on the high-order design of pneumatic servo system is still a challenging topic.

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Fig. 7. Experimental results of pneumatic cylinder with control law (29), 8 = 2, M= 30 kg for 0 < t-< 3 and M = 100 kg for 3 < t-< 6: (a) the output tracking of pneumatic cylinder; (b) the output tracking error of pneumatic cylinder; (c) the input voltage of the power amplifier.

Acknowledgements--The authors are grateful to Dr Zongli Lin, an assistant Professor of State University of New York, for pointing out mistakes in the proof of Theorem 3.1 of this paper and for providing some valuable references for this work. The authors would also like to thank the SMC Co. Ltd for supporting this work.

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(Received 7 October 1996)